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You searched for subject:(Angle action variables). Showing records 1 – 3 of 3 total matches.

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University of Kansas

1. Sofiani, Majed. KAM Stability of The Kepler Problem with a General Relativistic Correction Term.

Degree: MA, Mathematics, 2017, University of Kansas

In this work, we will be investigating a specific Hamiltonian system, namely, the Kepler problem with a correction term \frac{δ}{r3} added to the potential energy. Our objective is to show that the system is stable in the sense of the KAM theorem. In the first sections, we introduce essential concepts and tools that will be used in the process of understanding and showing how the KAM theorem works with our system. These concepts and tools are: Hamiltonian formalism, canonical transformations, the Hamilton-Jacobi equation and Action-Angle variables. In the last section, we state the KAM theorem and, based on the results we obtain from previous sections, we can conclude that the system is in fact stable in the sense of the KAM theorem. An informal statement of the KAM theorem is that if the unperturbed Hamiltonian system H0, expressed in the action variable J, is non-degenerate, then under sufficiently small perturbation ε H1 we have that \begin{equation*}\label{pert} H(J,Φ)=H0(J)+ε H1(J,Φ) \end{equation*} for ε 0. Advisors/Committee Members: Schaad, Beat (advisor), Stefanov, Atanas (cmtemember), Stanislavova, Milena (cmtemember).

Subjects/Keywords: Mathematics; Action-Angle variables; General relativity; KAM theorem; Kepler Problem

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APA (6th Edition):

Sofiani, M. (2017). KAM Stability of The Kepler Problem with a General Relativistic Correction Term. (Masters Thesis). University of Kansas. Retrieved from http://hdl.handle.net/1808/25828

Chicago Manual of Style (16th Edition):

Sofiani, Majed. “KAM Stability of The Kepler Problem with a General Relativistic Correction Term.” 2017. Masters Thesis, University of Kansas. Accessed November 26, 2020. http://hdl.handle.net/1808/25828.

MLA Handbook (7th Edition):

Sofiani, Majed. “KAM Stability of The Kepler Problem with a General Relativistic Correction Term.” 2017. Web. 26 Nov 2020.

Vancouver:

Sofiani M. KAM Stability of The Kepler Problem with a General Relativistic Correction Term. [Internet] [Masters thesis]. University of Kansas; 2017. [cited 2020 Nov 26]. Available from: http://hdl.handle.net/1808/25828.

Council of Science Editors:

Sofiani M. KAM Stability of The Kepler Problem with a General Relativistic Correction Term. [Masters Thesis]. University of Kansas; 2017. Available from: http://hdl.handle.net/1808/25828


University of Texas – Austin

2. Li, Meng, 1991-. Alfvén modes and wave-particle interaction in a tokamak.

Degree: PhD, Physics, 2017, University of Texas – Austin

This work is motivated by the nonlinear wave-particle interaction problems. To build a self-consistent theory, we consider eigenmodes of the bulk plasma as well as the dynamics of the energetic particles. The modes of our particular interest are the Alfvén Cascades and the Toroidicity Alfven Eigenmodes (TAE), which we describe using Magnetohydrodynamic(MHD) analysis and the AEGIS codes. We investigate the stabilizing effect for the Alfvenic waves from continuum damping, especially near the TAE gap. For the kinetic description of the energetic particles, we propose new canonical straight field line coordinates to model the guiding center motion. We then formulate wave-particle interaction problem using the action-angle variables. In Chapter 2, we interpret Alfvén Cascades observed in Madison Symmetric Torus (MST). We do linear MHD calculations and find the mode frequency, structure, and stability boundary. We then perform MHD simulation using the AEGIS code, with the equilibrium reconstructed from experiment. The result is discussed and compared with the experimentally observed features. In Chapter 3, we analyze continuum damping for Alfvénic waves, especially in the extreme situation near the TAE gap. We find that the continuum tip absorption feature is actually related to the existing of TAEs in the gap. On the technical level, we improve the numerical scheme of AEGIS and resolve two closely-spaced singularities. As a result, the absorption features observed in the simulation show good agreement with our analytical calculation. In order to simulate the energetic particle guiding center motion in the Hamiltonian form, we propose a new set of straight magnetic field line coordinates. The new coordinates exist for general tokamak devices and facilitate both MHD calculations and energetic particles. The new coordinate system makes it very convenient to take the advantage of the Hamiltionian structure of the guiding center motion. We use a canonical transformation to action-angle variables to formulate the interaction model for particles. The action-angle variables allow us to resolve wave-particle resonances and describe the conserved quantities for resonance particles. The model can give us a complete picture for nonlinear stage of wave-particle interaction. Advisors/Committee Members: Breizman, Boris N. (advisor), Berk, Herbert (committee member), Morrison, Philip (committee member), Fitzpatrick, Richard (committee member), Gamba, Irene (committee member).

Subjects/Keywords: Magnetic confinement fusion; Wave-particle interaction; Magnetohydrodynamics; Aflven modes; Action-angle variables

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APA (6th Edition):

Li, Meng, 1. (2017). Alfvén modes and wave-particle interaction in a tokamak. (Doctoral Dissertation). University of Texas – Austin. Retrieved from http://hdl.handle.net/2152/63061

Chicago Manual of Style (16th Edition):

Li, Meng, 1991-. “Alfvén modes and wave-particle interaction in a tokamak.” 2017. Doctoral Dissertation, University of Texas – Austin. Accessed November 26, 2020. http://hdl.handle.net/2152/63061.

MLA Handbook (7th Edition):

Li, Meng, 1991-. “Alfvén modes and wave-particle interaction in a tokamak.” 2017. Web. 26 Nov 2020.

Vancouver:

Li, Meng 1. Alfvén modes and wave-particle interaction in a tokamak. [Internet] [Doctoral dissertation]. University of Texas – Austin; 2017. [cited 2020 Nov 26]. Available from: http://hdl.handle.net/2152/63061.

Council of Science Editors:

Li, Meng 1. Alfvén modes and wave-particle interaction in a tokamak. [Doctoral Dissertation]. University of Texas – Austin; 2017. Available from: http://hdl.handle.net/2152/63061

3. Horsin, Romain. Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics.

Degree: Docteur es, Mathématiques et Applications, 2017, Rennes 1

Cette thèse porte sur le comportement en temps long de solutions d’équations de type Vlasov, principalement le modèle Vlasov-HMF. On s’intéresse en particulier au phénomène d’amortissement Landau, prouvé mathématiquement dans divers cadres, pour plusieurs équations de type Vlasov, comme l’équation de Vlasov-Poisson ou le modèle Vlasov-HMF, et présentant certaines analogies avec le phénomène d’amortissement non visqueux pour l’équation d’Euler 2D. Les résultats qui y sont décrits sont les suivants. Le premier est un théorème d’amortissement Landau pour des solutions numériques du modèle Vlasov-HMF, obtenues par discrétisation en temps de ce dernier via des méthodes de splitting. Nous prouvons en outre la convergence des schémas numériques. Le second est un théorème d’amortissment Landau pour des solutions du modéle Vlasov-HMF linéarisé autour d’états stationnaires inhomogènes. Ce théorème est accompagné de nombreuses simulations numériques destinées à étudier numériquement le cas non-linéaire, et semblant mettre en lumière de nouveaux phénomènes. Enfin, le dernier résultat porte sur la discrétisation en temps de l’équation d’Euler 2D par un intégrateur de Crouch-Grossman symplectique. Nous prouvons la convergence du schéma.

This thesis concerns the long time behavior of certain Vlasov equations, mainly the Vlasov- HMF model. We are in particular interested in the celebrated phenomenon of Landau damp- ing, proved mathematically in various frameworks, foar several Vlasov equations, such as the Vlasov-Poisson equation or the Vlasov-HMF model, and exhibiting certain analogies with the inviscid damping phenomenon for the 2D Euler equation. The results described in the document are the following.The first one is a Landau damping theorem for numerical solutions of the Vlasov-HMF model, constructed by means of time-discretizations by splitting methods. We prove more- over the convergence of the schemes. The second result is a Landau damping theorem for solutions of the Vlasov-HMF model linearized around inhomogeneous stationary states. We provide moreover a quite large amount of numerical simulations, which are designed to study numerically the nonlinear case, and which seem to show new phenomenons. The last result is the convergence of a scheme that discretizes in time the 2D Euler equation by means of a symplectic Crouch-Grossmann integrator.

Advisors/Committee Members: Faou, Erwan (thesis director), Rousset, Frédéric (thesis director).

Subjects/Keywords: Équations de type Vlasov; Équations d’Euler; Équations de transport; Amortissement Landau; État stationnaire; Méthodes de splitting; Méthodes semi-Lagrangiennes; Intégrateur symplectique; Intégrateur de Crouch-Grossman; Analyse d’erreur rétrograde; Systèmes hamiltoniens; Coordonnées action-angle; Vlasov equations; Euler equations; Transport equations; Landau damping; Stationary state; Splitting methods; Semi-Lagrangian methods; Symplectic integrator; Crouch-Grossman integrator; Backward error analysis; Hamiltonian systems; Angle-action variables

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Horsin, R. (2017). Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics. (Doctoral Dissertation). Rennes 1. Retrieved from http://www.theses.fr/2017REN1S062

Chicago Manual of Style (16th Edition):

Horsin, Romain. “Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics.” 2017. Doctoral Dissertation, Rennes 1. Accessed November 26, 2020. http://www.theses.fr/2017REN1S062.

MLA Handbook (7th Edition):

Horsin, Romain. “Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics.” 2017. Web. 26 Nov 2020.

Vancouver:

Horsin R. Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics. [Internet] [Doctoral dissertation]. Rennes 1; 2017. [cited 2020 Nov 26]. Available from: http://www.theses.fr/2017REN1S062.

Council of Science Editors:

Horsin R. Comportement en temps long d'équations de type Vlasov : études mathématiques et numériques : Long time behavior of certain Vlasov equations : mathematics and numerics. [Doctoral Dissertation]. Rennes 1; 2017. Available from: http://www.theses.fr/2017REN1S062

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